Tensor[HodgeStar] - apply the Hodge star operator to a differential form
Calling Sequences
HodgeStar(g, omega)
Parameters
g - a metric tensor
omega - a differential form
option - (optional) the keyword argument detmetric
Examples
withDifferentialGeometry:withTensor:
Example 1.
First create a 5-dimensional manifold M and define a metric tensor g on the tangent space of M.
DGsetupx1,x2,x3,x4,x5,M1:
g≔evalDGdx1&tdx1+dx2&tdx2+dx3&tdx3+dx4&tdx4+dx5&tdx5
g:=dx1dx1+dx2dx2+dx3dx3+dx4dx4+dx5dx5
The standard basis dx1,dx2, ...,dx5 is an orthonormal basis for g and therefore the Hodge star is easily computed.
HodgeStarg,dx1
dx2⋀dx3⋀dx4⋀dx5
HodgeStarg,dx2
−dx1⋀dx3⋀dx4⋀dx5
HodgeStarg,dx2&wdx3
dx1⋀dx4⋀dx5
HodgeStarg,dx2&wdx4
−dx1⋀dx3⋀dx5
HodgeStarg,dx2&wdx3&wdx4
−dx1⋀dx5
Example 2.
To show the dependence of the Hodge star upon the metric, we consider a general metric g on a 2-dimensional manifold.
DGsetupx,y,M2:
g≔evalDGadx&tdx+bdx&tdy+dy&tdx+cdy&tdy
g:=adxdx+bdxdy+bdydx+cdydy
HodgeStarg,dx
1ac−b2bdx+1ac−b2cdy
HodgeStarg,dy
−1ac−b2adx−1ac−b2bdy
f≔HodgeStarg,dx&wdy
f:=1ac−b2
HodgeStarg,f
dx⋀dy
Example 3.
The Laplacian of a function with respect to a metric g can be calculated using the exterior derivative operation and the Hodge star operator.
To illustrate this result, we use the Euclidean metric in polar coordinates r,ϑ.
DGsetupr,θ,M3:
g≔evalDGdr&tdr+r2dtheta&tdtheta
g:=drdr+r2dthetadtheta
To simplify the definition of the Laplacian, we define the Hodge operator with g fixed.
Hodge≔f↦HodgeStarg,fassuming0<r
Hodge:=f→DifferentialGeometry:-Tensor:-HodgeStarg,fassuming0<r
To display the Laplacian of φ in compact form we invoke the PDEtools[declare] command.
PDEtoolsdeclareφr,θ
φr,θwill now be displayed asφ
Here is the formula for the Laplacian in terms of HodgeStar and ExteriorDerivative. Recall that @ is the composition of functions.
Δ≔Hodge@ExteriorDerivative@Hodge@ExteriorDerivativeφr,θ
Δ:=rφr+r2φr,r+φθ,θr2
Example 4.
The HodgeStar program also works in the more general context of a vector bundle E→M.
DGsetupx,y,u,v,w,E
frame name: E
g≔evalDGdu&tdu+dv&tdv+dw&tdw
g:=dudu+dvdv+dwdw
HodgeStarg,du&wdv−3du&wdw+2dv&wdw
2du+3dv+dw
Example 5.
The HodgeStar operation can also be performed using an indefinite metric. The keyword argument detmetric = -1 must be used when the metric has negative determinant.
DGsetupx1,x2,x3,x4,M5:
g≔evalDGdx1&tdx1+dx2&tdx2+dx3&tdx3−dx4&tdx4
g:=dx1dx1+dx2dx2+dx3dx3−dx4dx4
HodgeStarg,dx1,detmetric=−1
dx2⋀dx3⋀dx4
HodgeStarg,dx3&wdx4,detmetric=−1
−dx1⋀dx2
Description
See Also
DifferentialGeometry
Tensor
DGinfo
ExteriorDerivative
MetricDensity
PermutationSymbol
RaiseLowerIndices
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